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Let $H$ be a Hopf algebra and $K$ a Hopf subalgebra of $H$. If $H$ is finite-dimensional, then by the Nichols-Zoeller Theorem $H$ is free as a left (and right) module over $K$.

Moreover, the same conclusion holds when $H$ is pointed, by the main theorem of [D. Radford: Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra 45, 266-273 (1977)], available here

http://www.sciencedirect.com/science/article/pii/002186937790326X

In both the previous cases, can one always find a basis $\mathcal{B}$ of $H$ over $K$ with $1\in \mathcal{B}$?

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In the pointed case, this should follow from the proof in the Radford's paper available at the quoted link.

In your notation, consider the left $K$-module $C=KG$, where $G$ is the group of group-like elements of $H$. It is easy to see (and proved in the paper) that a basis of $C$ over $K$ is given by any set of right coset representatives for the subgroup of $G$ consisting of the group-like elements of $K$.

Now, in order to prove that $H$ is free as left $K$-module Radford shows that there exists a certain series

$$ 0=C^{(-1)}\subset C^{(0)} \subset C^{(1)} \cdots \subset C^{(\infty)}=\bigcup_{n\geq -1}C^{(n)} =H $$

with $C^{(0)}=C$ such that $C^{(n)}/C^{(n-1)}$ is a free left $K$-module for any $n\geq 0$.

How to find a basis with the desired property is then clear.

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